Geometry
# 3D Coordinate Geometry

The points $A$, $B$, $P$, and $Q$ all lie on one line, with $A=(-3, 5, 8).$

Point $P$ lies in the $xy$-plane between $A$ and $B$ such that the distance from $A$ to $P$ is twice the distance from $P$ to $B$. Furthermore, $Q$ sits on the $z$-axis such that the distance from $A$ to $Q$ is twice the distance from $Q$ to $B$.

If $B=(x,y,z),$ what is the value of $x+y+z?$

An infinite column is centered along the $z$-axis. It has a square cross-section of side length 10. It is cut by the plane $4x - 7y + 4z = 25.$

What is the area of the surface cut?